MinT - Best Known (19, 19+∞, s)-Nets in Base 4

Best Known (19, 19+∞, s)-Nets in Base 4

(19, 19+∞, 33)-Net over F₄ — Constructive and digital

Digital (19, m, 33)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (19, 32)-sequence over F₄, using
- t-expansion [i] based on digital (15, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(19, 19+∞, 41)-Net over F₄ — Digital

Digital (19, m, 41)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (19, 40)-sequence over F₄, using
- t-expansion [i] based on digital (18, 40)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
    - Drinfeld modules of rank 1 [i]

(19, 19+∞, 69)-Net in Base 4 — Upper bound on s

There is no (19, m, 70)-net in base 4 for arbitrarily large m, because

m-reduction [i] would yield (19, 206, 70)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²⁰⁶, 70, S₄, 3, 187), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 655767 521039 926610 160299 985568 508871 509777 445500 566969 530371 604912 210544 906420 855058 587696 756197 562568 810482 328823 171437 821952 / 47 > 4²⁰⁶ [i]